Problem 29 - College Trigonometry

Question

In Exercises \(25-39\), find a parametric description for the given oriented curve. the curve \(x=y^{2}-9\) from \((-5,-2)\) to \((0,3)\).

Step-by-Step Solution

Answer

The parametric description is \(x(t) = t^2 - 9\), \(y(t) = t\) for \(-2 \leq t \leq 3\).

Step 1: Understand the Problem

We need to find a parametric description of the curve defined by the equation \(x = y^2 - 9\) for \(y\) values between -2 and 3. This curve is a parabola opening to the right.

Step 2: Assign a Parameter to the Curve

To parametrize the curve, we choose a parameter \(t\) such that it varies linearly with \(y\). Let \(y = t\), where \(t\) ranges from -2 to 3, consistent with the endpoints of the given curve.

Step 3: Express x in Terms of the Parameter

Substitute \(y = t\) into the equation of the curve. This gives us \(x = t^2 - 9\). Now both \(x\) and \(y\) are expressed in terms of the parameter \(t\).

Step 4: Write the Parametric Equations

From Steps 2 and 3, the parametric equations for the curve are \(x(t) = t^2 - 9\) and \(y(t) = t\). These parametric equations represent the curve as \(t\) ranges from -2 to 3.

Step 5: Verify the Parametric Equations

Check that the starting point (-5,-2) corresponds to \(t = -2\) and the ending point (0,3) corresponds to \(t = 3\). When \(t = -2\), \(x = (-2)^2 - 9 = -5\) and \(y = -2\), which is the starting point. When \(t = 3\), \(x = 3^2 - 9 = 0\) and \(y = 3\), which is the ending point.

Key Concepts

Parabola

A parabola is a special type of curve that you might recognize from its familiar U-shape in mathematics. It is defined as the set of all points in a plane that are equidistant from a fixed point, called the focus, and a fixed line, called the directrix. In most coordinate planes, a standard form of a parabolic equation is given as either

• Vertical parabolas: \(y = ax^2 + bx + c\)
• Horizontal parabolas: \(x = ay^2 + by + c\)

In this exercise, we are dealing with a horizontal parabola, where the equation is given by \(x = y^2 - 9\). This equation tells us that the parabola opens to the right because the variable \(x\) is expressed in terms of \(y^2\). By plugging in different values of \(y\) into the equation, we can determine the corresponding values of \(x\), which help in plotting the parabolic curve accurately. By understanding these properties, you'll see how parabolas are the building block for solving complex problems involving curves.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. This system involves placing geometric figures into the Cartesian plane created by the intersection of two number lines—the x-axis and the y-axis. This branch of geometry allows for the precise representation of figures by using equations.Coordinates provide the position of any point on the plane, formatted as \((x, y)\). Relationships between these points can often be described using algebraic expressions, like equations of lines (e.g., \(y = mx + c\)), and, as in our problem, the equation of a parabola \(x = y^2 - 9\).In our exercise, coordinate geometry aids in visualizing the parabola's path or oriented curve from one specified point to another. By converting the parabola's equation into parametric form, the geometry becomes even more intuitive, making it possible to trace or sketch the curve piece by piece between the given points \((-5, -2)\) and \((0, 3)\).

Curve Parametrization

Curve parametrization is a powerful tool in math that assigns each point along a curve to a unique parameter. This method simplifies the complex task of describing curves by using one or more parameters that "animate" the curve, showing how it develops.In our exercise, we have chosen the parameter \(t\) to describe the curve. We set \(y = t\), allowing \(t\) to vary from -2 to 3. This choice reflects the span of the original points \((-5,-2)\) to \((0,3)\). As \(t\) changes, it provides a direct one-to-one mapping of the \(y\) values, and subsequently, we use it to express \(x\) as well, like so: \(x(t) = t^2 - 9\) and \(y(t) = t\).With parametric equations, we can trace, plot, and analyze curves more easily. It shows the path or trajectory of the motion along the parabolic curve. Parametrization is not only limited to parabolas but can be extended to other curves like ellipses and hyperbolas, offering significant flexibility in mathematical modeling and analysis.